What is 8/2(2+2)? I believe it's 16 but my friends say it's 1.

[Hyper_Deity]

[HELP] I Need an Answer to This:

What is 8/2(2+2)?

I believe it's 16 but my friends say it's 1.

 

[mmm4444bot]

What is 8/2(2+2)? I believe it's 16 but my friends say it's 1.

You are correct; it's 16.

We do multiplication and division in the order they occur, reading from left to right.

8/2 is done first; the multiplication by 4 is done second.

Your friends are thinking of the following:

8/[2(2+2)]

The numerator is red and the denominator is blue.

Cheers :cool:

 

[Harry_the_cat]

What is 8/2(2+2)? I believe it's 16 but my friends say it's 1.

If it is written as \(\displaystyle \frac {8}{2(2+2)}\) then the answer is 1.

 

[tkhunny]

8/2(2+2)

In my primary programming language, everything is interpreted strictly right-to-left.

8/2(2+2) = 8/2*(4) = 8/8 = 1

In standard U.S. Order of Operations (Other places may teach the order differently.)

8/2(2+2) = (8/2)*(2+2) = 4*4= 16

In other words, there is no substitute for knowing what you are doing. There is no shame in adding additional parentheses for clarity.

Personally, I would avoid implied operations whenever possible.


Also, Harry_the_Cat brings up a very important point. Textbook print and in-line print are not usually interpreted the same. For one, the vinculum is an effective grouping symbol AND a sign of division in a textbook. The in-line slash is only division. It groups nothing.

 

[ksdhart2]

Just to throw my hat in the ring, part of the confusion surrounding PEMDAS/BODMAS probably stems from the fact that different calculators sometimes give different results for the exact same input. Case in point, the calculator program built in to Windows 10. In Standard mode, it always processes operations left-to-right, storing each answer in its memory to apply to the next operator. As such, it completely ignores the order of operations. However, if you're in Scientific mode, the order of operations is preserved.

Screenshot of Standard mode
Screenshot of Scientific mode

 

[stapel]

What is 8/2(2+2)?

I believe it's 16 but my friends say it's 1.

This is the one issue with the rules for the Order of Operations. Opinion is often fairly evenly split regarding whether the parenthetical is multiplied after the processing of the division, so that the posted expression means the following:

. . . . .\(\displaystyle \dfrac{8}{2}\, \times\, (2\, +\, 2)\)

...or whether the parenthetical is somehow "bound" to the preceding 2, so that the expression really means the following:

. . . . .\(\displaystyle \dfrac{8}{2\, (2\, +\, 2)}\)

Had the expression been the following:

. . . . .8/x(2+2), with x = 2

...or:

. . . . .8/4x, with x = 2

...then I suspect more would swing toward the second interpretation, as "multiplication by juxtaposition" will be instinctively viewed as being more "strong". (here) But this is an instinct, not a rule.

However, this instinct generally being fairly strong in math and science folks, this debate will likely continue to rage, no matter how many times this exact sort of expression is posted, whether on forums, Facebook, Reddit, or elsewhere.

 

[mmm4444bot]

… Opinion is often fairly evenly split regarding whether the parenthetical is multiplied after the processing of the division [or before] …

Setting aside context and typesetting formats momentarily, may we say that the reason for an unambiguous definition of the Order of Operations is so that people do not need to rely on opinions?

If all we have is text characters 1/4x, then I believe the definition is clear. That's 1/4th of x.

If, on the other hand, we have something like the following, then I take pause.

1 / 4x

2/8x

With the former, I assume they intentionally placed spaces around the division operator to imply that 4x is the denominator. Yet, I'll probably ask for confirmation.

With the latter, I'll wonder why the coefficient was not reduced to lowest terms, and then I'll (probably) consider that maybe 8x is the denominator. I'd better ask, anyway.

But, without any context, the character string 1/4x means x/4, to me, because that's what the definition states. :cool:

 

[stapel]

Setting aside context and typesetting formats momentarily, may we say that the reason for an unambiguous definition of the Order of Operations is so that people do not need to rely on opinions?

I know, right? But still, this one type of expression continually crops up, and always generates more heat than light. Oh, well.

 

[mmm4444bot]

I know, right? But still, …

Perhaps, the definition needs to be stressed more in class.

An issue I see often is when a student immediately moves to distribute because "we do parentheses first". They misunderstand the definition.

 

[tkhunny]

"unambiguous" requires sound understanding. I recall a class of 30 college juniors of mathematics. TWO very long problems with ONLY simple arithmetic. Only two of us managed both. It's a tricky business, no matter how we try to claim that it is clear. Use more parentheses and ASK if ANYONE could possibly misconstrue. It is hard enough to write so that anyone can understand. The challenge is to write so that no one can misunderstand. Good luck to us all on that last one.

 

[Harry_the_cat]

This is the one issue with the rules for the Order of Operations. Opinion is often fairly evenly split regarding whether the parenthetical is multiplied after the processing of the division, so that the posted expression means the following:

. . . . .\(\displaystyle \dfrac{8}{2}\, \times\, (2\, +\, 2)\)

...or whether the parenthetical is somehow "bound" to the preceding 2, so that the expression really means the following:

. . . . .\(\displaystyle \dfrac{8}{2\, (2\, +\, 2)}\)

Had the expression been the following:

. . . . .8/x(2+2), with x = 2

...or:

. . . . .8/4x, with x = 2

...then I suspect more would swing toward the second interpretation, as "multiplication by juxtaposition" will be instinctively viewed as being more "strong". (here) But this is an instinct, not a rule.

However, this instinct generally being fairly strong in math and science folks, this debate will likely continue to rage, no matter how many times this exact sort of expression is posted, whether on forums, Facebook, Reddit, or elsewhere.

Interesting observation. And probably one that needs to be addressed somewhere and incorporated into the teaching of BOMDAS (or whatever acronym you use).

So does 8/2(2+2) = 8/2x(2+2) ?. The absence of the multiplication sign (because it looks like an "x") on the LHS causes the confusion.

No doubt the RHS = 16.

Does the absence of the multiplication sign imply extra brackets ie does 8/2(2+2) imply 8/[2(2+2)] in a similar way to the use of the vinculum in \(\displaystyle \frac {8}{2(2+2)}\) does? Interesting!